Gaia FGK Benchmark Stars: fundamental T e ff and log g of the third version

Context. Large spectroscopic surveys devoted to the study of the Milky Way, including Gaia, use automated pipelines to massively determine the atmospheric parameters of millions of stars. The Gaia FGK Benchmark Stars are reference stars with T e ff and log g derived through fundamental relations, independently of spectroscopy, to be used as anchors for the parameter scale. The first and second versions of the sample have been extensively used for that purpose, and more generally to help constrain stellar models. Aims. We provide the third version of the Gaia FGK Benchmark Stars, an extended set intended to improve the calibration of spectroscopic surveys, and their interconnection. Methods. We have compiled about 200 candidates which have precise measurements of angular diameters and parallaxes. We determined their bolometric fluxes by fitting their spectral energy distribution. Masses were determined using two sets of stellar evolution models. In a companion paper we describe the determination of metallicities and detailed abundances. Results. We provide a new set of 192 Gaia FGK Benchmark Stars with their fundamental T e ff and log g , and with uncertainties lower than 2% for most stars. Compared to the previous versions, the homogeneity and accuracy of the fundamental parameters are significantly improved thanks to the high quality of the Gaia photometric and astrometric data.


Introduction 1
The last decade has been marked by a large observational ef-  (Creevey et al. 2023;Fouesneau et al. 2023).

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In particular, two datasets were released that mainly include F-, 10 G-, and K-type stars, one for 5.6 million stars with APs based on 11 medium resolution spectra from the Radial Velocity Spectrome-12 ter (Recio-Blanco et al. 2023), and the other one for 471 million The full catalogue is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/?/? stars with APs based on low-resolution spectra from the blue and red prisms, parallax, and integrated photometry (Andrae et al. 2023).The methodologies used for the massive determination of atmospheric parameters rely on stellar models that are not perfect and not able to reproduce real spectra exactly, causing some biases that have to be corrected.
The Gaia FGK benchmark stars (GBS) are reference stars to be used for the calibration and the validation of spectroscopic methods of parametrisation.They are chosen to cover the range of F, G, and K spectral types at different luminosities and metallicities, and to have the necessary observations available to determine their effective temperature and surface gravity independently from spectroscopy, at a precision level of 1-2%.The determination of T eff and log g is performed through the fundamental relations implying observable quantities (angular diameters directly measured by interferometry, bolometric fluxes, and paral-laxes) and the mass, which is the only parameter that depends on theoretical assumptions.
The first and second versions of the GBS (hereafter V1 and V2, respectively) were presented in a series of papers.Heiter et al. (2015), hereafter Paper I, describe the initial selection of 34 stars, including the Sun, and the determination of their fundamental effective temperatures and surface gravities, resulting in the GBS V1 sample.Blanco-Cuaresma et al. (2014) (Paper II) present the library of high-resolution spectra that was assembled and used to determine metallicities (Jofré et al. 2014, Paper III) and elemental abundances of α−capture and iron-peak elements (Jofré et al. 2015, Paper IV).One limitation of the V1 sample was the small number of targets, in particular in the metal-poor regime.Metal-poor stars are usually distant and faint, which makes them difficult to observe in interferometry.In Paper V, Hawkins et al. (2016) proposed a list of ten metal-poor stars to be included in the GBS sample.The GBS V2 sample summarised by Jofré et al. (2018) includes 36 stars, merged from Paper I and Paper V. The change in number from 34 to 36 comes from the addition of five metal-poor stars from Paper V and the removal of some stars from Paper I because their spectroscopic analysis indicated that they could not be recommended as reference stars.
However, V2 was an intermediate version where the fundamental properties of the stars were not redetermined owing to the lack of direct and accurate measurements of angular diameters for some stars.
The material provided in these series of papers consists of accurate APs for stars covering an extensive range of spectral types and metallicities, in addition to a library of high-resolution and high signal-to-noise spectra from which line-by-line abundances are also provided.This material can be further exploited in spectroscopic studies.Paper VI of the GBS series (Jofré et al. 2017) does indeed report on a collective work using the GBS to investigate the different sources of uncertainties in elemental abundances in order to improve spectroscopic pipelines.
The ultimate goal of the efforts dealing with GBS is to provide the fundamental T eff and log g scales and an external reference for abundances to spectroscopic surveys.Despite their limitation in sample size and parallax precision previous to Gaia data, the GBS have been extensively used in the past years.The Gaia astrophysical parameters' inference system (Bailer-Jones et al. 2013;Creevey et al. 2023) made use of GBS for the validation of the stellar parameters published in Gaia DR3.The GBS are also a fundamental source of calibration and validation of the Gaia-ESO survey (Gilmore et al. 2022;Randich et al. 2022;Hourihane et al. 2023), of the RAVE survey (Steinmetz et al. 2020b,a), and of the GALAH survey (Buder et al. 2021).The OCCASO project (Casamiquela et al. 2019) has systematically observed two GBS giants, Arcturus and µ Leo, to validate chemical abundances of open clusters.Upcoming large projects such as WEAVE (Jin et al. 2022) are also making use of the GBS.
Calibrations based on GBS can help to make surveys more homogeneous and mutually compatible so that they can be combined into the most comprehensive database of chemical measurements for the study of the Milky Way stellar populations (Jofré et al. 2018).The applications of the GBS, however, can extend far beyond this specific purpose.As for the study presented in Paper VI (see also Blanco-Cuaresma 2019), many spectroscopic studies have benefited from the GBS effort.For example, Adibekyan et al. (2020) used the GBS to assess the performances of the ESPRESSO, PEPSI, and HARPS high-resolution spectrographs, while Heiter et al. (2021) used some spectra from Paper II to assess the quality of hundreds of spectral lines and the corresponding atomic and molecular data used for the abundance analyses of FGK-type stars carried out within the Gaia-ESO sur-94 vey (see also Kondo et al. 2019;Fukue et al. 2021, for lines in 95 the Infrared).Amarsi et al. (2022) and Lind et al. (2022) used 96 the GBS to quantify the differences in abundances derived using 97 state-of-the-art 3D non local thermodynamic equilibrium (LTE) 98 atmosphere models and the standard 1D LTE models.

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In addition to spectroscopy, the GBS help to constrain better 100 stellar evolution models.For example, Sahlholdt et al. (2019) 101 determined ages of the GBS as a way to test the reliability of 102 the determination of stellar ages for various stellar populations; 103 Serenelli et al. (2017) used GBS to validate their asteroseismic 104 analysis performed on dwarfs and subgiants.The GBS have also 105 been used as a validation for the PLATO stellar analysis pipeline 106 (Gent et al. 2022).Many of the lessons learnt from the GBS are 107 further discussed in Jofré et al. (2019).
108 However, we are aware that the current sample of GBS is 109 still imperfect and too small to make a satisfactory interconnec-110 tion between surveys.This is why an extension of the sample is 111 required.The V1 and V2 GBS samples were also limited by the 112 parallax accuracy needed for a fundamental log g determination.113 This is not an issue anymore thanks to the exquisite astrometric 114 quality of the Gaia data (Gaia Collaboration et al. 2016).

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In this Paper VII of the series, we present the extended sam-116 ple and third version of the GBS (GBS V3) that includes about 117 200 stars.We took advantage of recent interferometric studies 118 that provided new measurements of angular diameters for large 119 samples of stars (e.g.Ligi et al. 2016;Baines et al. 2018Baines et al. , 2021;;120 van Belle et al. 2021) and for metal-poor stars (e.g.Creevey et al. 121 2015; Karovicova et al. 2018Karovicova et al. , 2020)).As explained in Sect.2, we 122 selected new GBS candidates based on quality criteria applied on 123 interferometric measurements.Sect. 3 describes the compilation 124 of angular diameters and fluxes that are needed to compute the 125 fundamental T eff .Bolometric fluxes (F bol ) were homogeneously 126 computed by the method of spectral energy distribution (SED) 127 fitting based on a large collection of (spectro)photometric data.128 Sect. 4 deals with the determination of log g with parallaxes from 129 Gaia DR3 (Gaia Collaboration et al. 2023c), or Hipparcos (van 130 Leeuwen 2007) for the brightest stars, and with masses inferred 131 from a state-of-the-art methodology and stellar tracks.At each 132 of these different steps, we assess the uncertainties of the stellar 133 parameters.Sect. 5 provides an overview of the sample proper-134 ties and shows some comparisons to T eff and log g from different 135 catalogues, before our concluding remarks in Sect.6.All of the 136 compiled and computed parameters of this work are given in the 137 form of a catalogue distributed by the CDS.We note that these 138 parameters still require a last iteration considering [Fe/H] values, 139 which are needed for the estimation of F bol and masses and have 140 been adopted from the literature for this work, and they are con-141 sistent with our fundamental parameters.This is a necessary step 142 to recommend our parameters for reference (Heiter et al. 2015).143 The accompanying Paper VIII  tion) presents homogeneous determinations of [Fe/H] and of de-145 tailed abundances of the GBS V3 derived from a spectroscopic 146 analysis.For this purpose a large dataset of high-resolution, high 147 signal-to-noise spectra was collected from public archives and 148 through our own observing programmes.The recommended pa-149 rameters and abundances of the GBS are appropriately updated 150 at the CDS.
high-quality measurement of angular diameter.Ideally we want our GBS sample to homogeneously cover the (T eff ,log g,[Fe/H]) space, which implies a special effort to add metal-poor stars.We have therefore searched the literature for GBS candidates fulfilling these criteria.
First, we considered the GBS from V1 and V2 (Heiter et al. 2015;Jofré et al. 2018).The GBS V1 sample has 29 FGK-type stars (including the Sun), four giants with T eff around 4000 K, corresponding to late K and early M spectral types, and one cooler M giant.The V2 sample was built from the V1 one, with the addition of metal-poor stars.For the V3 list we considered all the 39 V1 and V2 stars including several stars with indirect determinations of angular diameters.For all we searched for new direct determinations of angular diameters as well as other data needed to update their fundamental T eff and log g.We added to this list eight metal-poor stars from Karovicova et al. (2020Karovicova et al. ( , 2022a)), not part of V1 and V2, and two targets recently observed with the CHARA interferometer (Creevey et al. in preparation).This sample of 49 stars was our initial set.
To further extend the GBS sample, we searched for new candidates observed in interferometry.We used the compilation from the Jean-Marie Mariotti Center (JMMC), the JMMC Measured Stellar Diameters Catalogue (JMDC, Duvert 2016).This catalogue, regularly updated, intends to be as exhaustive as possible in listing all the measurements of stellar apparent diameters made with direct techniques.It is therefore a very appropriate resource to extend the GBS sample.The JMDC is a bibliographical catalogue which implies that some stars have multiple entries, resulting from studies with different instruments, in different bands and with different precisions.Deciding which value of angular diameter is the most appropriate for a given star can be challenging, in particular owing to non-homogeneous uncertainties listed in the JMDC.In addition, there are many stars in the JMDC which are not appropriate for our purpose, such as some classes of variable stars, hot stars, spectroscopic binaries, and fast rotators.In addition, some very uncertain measurements of angular diameters could propagate large uncertainties to T eff and should be discarded.Therefore we made a first selection to reject stars and measurements not relevant for our purpose.
To do so, we followed Salsi et al. (2020) who established accurate surface brightness-colour relations for different spectral types and luminosity classes.They applied three types of rejection criteria on the JMDC data.First they examined the stellar characteristics to reject variable and semi-regular pulsating stars, spectroscopic binaries and other multiple stars, fast rotators and stars with a doubtful luminosity class.Second, they used criteria on the quality of the interferometric measurements that we apply similarly (see Sect. 3.1).The third type of criterion is based on the uncertainty of the K magnitude.We considered their list of 106 carefully selected F5 to K7 dwarfs and giants that we added to the initial set (five stars were already there).However, the study of Salsi et al. (2020) does not take into account the metallicity of the stars since their objective is to infer radii of stars and planets in the context of the PLATO mission which mainly focuses on solar-like stars.For us the metallicity is essential since the GBS should be representative of all the Milky Way stellar populations.We aim to improve the sampling of the GBS in T eff and log g but also in [Fe/H] with as many GBS candidates as possible on the metal-poor side.We noticed that the criteria used by Salsi et al. (2020), in particular the photo- We then searched for additional stars in the September 2021 220 version of JMDC available at the CDS which includes 2013 mea-221 surements of 1062 stars, a significant increase compared to the 222 February 2020 version used by Salsi et al. (2020).In order to 223 find stars in the appropriate range of atmospheric parameters, 224 we used the PASTEL catalogue (Soubiran et al. 2016) and its 225 recent version which provides mean atmospheric parameters for 226 14 181 FGK stars (Soubiran et al. 2022).We expect PASTEL to 227 be complete for metal-poor stars brighter than V∼8.25, which is 228 the limiting magnitude of FGK-type stars with an interferomet-229 ric measurement in JMDC.Among the ∼500 stars in common 230 between PASTEL and JMDC, we considered 63 additional stars 231 to include in our sample, because they fill gaps in the AP space, 232 and their interferometric angular diameters fulfil the criteria of 233 Salsi et al. (2020).

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The resulting list of selected candidates for GBS V3 includes 235 201 stars (the Sun is not considered here) They are all mem-236 bers of the Hipparcos catalogue (ESA 1997) and only the ten 237 brightest ones are missing in Gaia DR3 (Gaia Collaboration et al. 238 2016).We keep the Sun in the GBS V3 since it is an obvious 239 benchmark star, although it is not observable in the same condi-240 tions as other stars.We do not discuss the Sun in the present pa-241 per, keeping its fundamental T eff and log g determined in Paper I 242 (we also note that a nominal value for the effective temperature 243 of the Sun was adopted at the XXIXth IAU General Assembly, 244 see Mamajek et al. 2015;Prša et al. 2016).

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In the following, metallicities [Fe/H] are needed for the de-246 termination of F bol from SEDs (to initialise the minimisation 247 process, see Sect.3.4), and for the determination of masses 248 from stellar evolutionary tracks (see Sect. 4.2).We have adopted 249 [Fe/H] from the literature for the 201 stars, mainly from the PAS-250 TEL catalogue.For a sake of homogeneity, we have not adopted 251 [Fe/H] from Papers III and V for stars in V1 and V2 because 252 they are corrected from non-local thermodynamic equilibrium 253 (NLTE) effects, while for all the other stars the literature values 254 are assuming LTE.It is the purpose of the forthcoming Paper 255 VIII to provide precise and homogeneous abundances of Fe and 256 other elements.This will imply some iterations to get the recom-257 mended T eff and log g of our targets.The luminosity L, the radius R, and the effective temperature 260 T eff of a given star are linked through the fundamental relation 261 L = 4πR 2 σT 4 eff , where σ is the Stefan-Boltzmann constant.The 262 fundamental relation can be expressed in a way that gives T eff 263 as a function of the limb-darkened angular diameter θ LD and the 264 bolometric flux F bol which are measurable quantities: where θ LD is in milliarcseconds (mas) and F bol in 266 10 −8 erg s −1 cm −2 or 10 −11 W m −2 .

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In the following subsections we describe our compilation of 268 measured angular diameters and fluxes.The fluxes were used to 269 compute F bol by means of SED fitting.Subsequently, Eq. (1) was 270 used to obtain T eff for the selected stars.As explained in Sect.2, the selection of GBS V3 stars was 273 mainly based on the JMDC which provides one or several val-274 ues of θ LD for each star.In particular, we considered the 106 275 targets that Salsi et al. (2020) used as calibration stars to define precise surface brightness-colour relations.Salsi et al. (2020) applied interferometric criteria to remove non reliable values of θ LD in the JMDC.They rejected measurements with a relative uncertainty on the angular diameter larger than 8%, and those based on observations in the 8-13 micron band or having a bad observation quality and/or a poor spatial frequency coverage in the visibility curve.They also rejected stars with inconsistent redundancies.We adopted their selected values of θ LD for the 106 stars.For HIP112748 and HIP54539, provided with two values of θ LD differing by less than 1%, we adopted the one with the lowest uncertainty.
For the remaining stars, we queried JMDC and the recent literature in order to retrieve the latest values of θ LD fulfilling the interferometric criteria applied by Salsi et al. (2020).When provided we inspected the visibility curves to evaluate the reliability of the measurement.
We found recent and precise θ LD for ten of the GBS V1 and V2.In particular, for three of the metal-poor benchmark stars new measurements are available, for HD103095 (HIP57939) and HD122563 (HIP68594) by Karovicova et al. (2020), and for HD140283 (HIP76976) by Karovicova et al. (2018).The two components of the binary α Cen were remeasured by Kervella et al. (2017), while four other targets were found in Baines et al. (2018Baines et al. ( , 2021).Among the six stars which had no direct θ LD in Paper I, only one (HIP48455, µ Leo) was observed in interferometry by Baines et al. (2018).Among the five metal-poor stars from Paper V with indirect values of θ LD , one (HIP92167) was observed by Karovicova et al. (2020).Thus, we are left with nine stars from V1 and V2 that are still without any direct measurement of θ LD .We keep them in a separate table for continuity of the samples, but we do not consider them as GBS anymore.
The final version of the GBS V3 includes 192 stars with a direct measurement of θ LD .For each we provide the limb darkened angular diameter with its uncertainty and the corresponding reference in Table A.1 of Appendix A and in the catalogue available at the CDS.The sample includes stars with small angular diameters such as HIP97527 (θ LD =0.231±0.006mas) and HIP93427 (θ LD =0.289±0.006mas), both of which are asteroseismic targets observed with the CHARA/PAVO instrument by Huber et al. (2012).The sample also includes Aldebaran (HIP21421) and Arcturus (HIP69673) which have angular diameters as large as ∼20 mas.The median angular diameter of the sample is 1.12 mas.
The relative θ LD uncertainties range from 0.1% (HIP87808) to 7% (HIP25993) with a median value of 1.1% (see histogram in Fig. 1).Two other stars have relative uncertainties larger than 5%, HIP7294 and HIP14838.In absolute values, the largest uncertainties occur for the two giants ψ Phe (HIP8837) and Arcturus (HIP69673), with θ LD =8.0±0.2 mas and θ LD =21.0±0.21mas, respectively.The two stars do not seem to have been reobserved recently, so that their θ LD is still that of Paper I.
We note that, for a small fraction of stars, we had to make a choice between the two or more values of θ LD fulfilling the adopted quality criteria.As shown in Fig. 2, several small diameters (typically θ LD <1.5 mas) disagree by more than 10%, but in general the agreement is at the 2σ level.We note three stars with estimations of their angular diameters differing by more than 3σ: HIP96441, HIP57939, HIP108870.
HIP96441 has three values of θ LD reported in the JMDC, that fulfil the interferometric criteria of Salsi et al. (2020): 0.861±0.015mas in the K band (Boyajian et al. 2012a), 0.753±0.009mas in the R band (White et al. 2013) and 0.749±0.007mas in the H band (Ligi et al. 2016).The first de- termination is not compatible with the two others, but Boyajian 340 et al. ( 2013) mention a calibration problem and discarded this 341 star.Between the two other values we adopted the most recent 342 one by Ligi et al. (2016).

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HIP57939 (HD103095) is a well-known metal-poor dwarf 344 studied by several authors.We adopted the latest determination, 345 θ LD =0.593±0.004mas, by Karovicova et al. (2020) who used 346 the combination of two instruments, VEGA and PAVO on the 347 CHARA interferometer giving a high confidence to their result.348 For HIP108870 we adopted the value θ LD =1.758±0.012mas 349 by Rains et al. (2020) which significantly differs from that pre-350 viously reported by Kervella et al. (2004), θ LD =1.89±0.02mas.351 Rains et al. (2020) have analysed this discrepancy, considering 352 that they obtained tighter constraints on the angular diameter by 353 better resolving the star, thanks to the configuration now avail-354 able at the VLTI.

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These cases of disagreement also illustrate the inhomogene-356 ity of uncertainties listed in JMDC, which sometimes only reflect 357 the precision of a fit, or also include systematic effects identi-358 fied at the calibration level.The dispersion among measurements 359 available for a given star is critical for small angular diameters, 360 typically below ∼1.5 mas, because it corresponds to discrepan-361 cies that can reach 10 to 15%.This illustrates the limitations 362 of measuring interferometric diameters in the sub-mas regime.363 Some inhomogeneity can also arise from different recipes ap-364 plied for the limb darkening correction.According to Eq. (1), a 365 variation of 10% in θ LD translates into a variation of 5% in T eff .366 Inversely, a 1% precision on T eff implies angular diameters ob-367 tained at the 2% level.

Compilation of magnitudes and fluxes
In order to build a SED for each star and measure the corresponding F bol we compiled fluxes using the VOSA tool1 (Bayo et al. 2008).VOSA allowed us to collect all the photometry available in the Virtual Observatory (VO) for our list of 201 stars (including the nine stars from V1 and V2 with an indirect θ LD ) and to convert magnitudes into fluxes thanks to an exhaustive description of all the existing filters.We only kept the photometry from the VO catalogues that contain at least fifty of our targets, namely 2MASS (Cutri et al. 2003), AKARI (Yamamura et al. 2010), Gaia DR3 (Gaia Collaboration et al. 2023c), GALEX (Bianchi et al. 2017), Strömgren photometric catalogues (Hauck & Mermilliod 1998;Paunzen 2015), Johnson UBV (Mermilliod 1987), IRAS (Neugebauer et al. 1984), Hipparcos (ESA 1997), Tycho-2 (Høg et al. 2000) and WISE (Cutri et al. 2021).
We note that the components of the bright binary star α Cen HIP103598 is a K4 giant with a metallicity 417 of −0.36 according to PASTEL, which has a well constrained 418 SED thanks to Gaia and Pulkovo spectrophotometry.HIP50564 419 is an F6 turn-off star with a metallicity of +0.10 according to 420 PASTEL, having only broad-band photometric observations.We 421 chose these stars to illustrate both the SED shape variations due 422 to different temperatures, and the more or less good coverage 423 of the SED depending on the availability of spectrophotometric 424 data.425

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The extinction towards each of the 201 targets was estimated 427 thanks to the recent 3D maps provided by Vergely et al. (2022), 428 based on the inversion of large spectroscopic and photometric 429 catalogues including Gaia DR3.We chose the closest map, cov-430 ering a volume of 3 kpc x 3 kpc x 800 pc at a resolution of 10 pc, 431 which is particularly well adapted for our sample of nearby stars.432 The extinction is low for most of the stars (90% of them have 433 A V < 0.05) which is not surprising owing to the small distances 434 of the GBS V3 from the Sun.Our GBS span distances from 3 pc 435 to 550 pc (deduced from parallaxes, see Sect.4.1).Five giants 436 have the highest extinction values, between 0.1 and 0.31 mag.437 As expected, A V is well correlated to the distance, as shown in 438 Although the observed fluxes compiled for the GBS cover a wide 441 range of wavelength, some extrapolation of the SED is needed to 442 integrate the full distribution and measure the total flux from the 443 star received at the Earth, F bol .To do so we followed the SED 444 fitting method previously used by Creevey et al. (2015) and Ligi 445 et al. ( 2016), based on the BASEL empirical library of spectra 446 (Lejeune et al. 1997)

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The BASEL library covers the following parameter ranges: < +1.0.It extends to 2 000 K for a subset of the log g and [M/H].

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The wavelength range spans 9.1 to 160 000 nm on a non-evenly ties using the same approach.These simulations were done 400 492 times where 400 was a balance between computing time and having a significant sample size (the results with 200 simulations were equivalent within the uncertainties and the standard deviation of the 400 simulations reproduced the uncertainties of the atmospheric parameters).For the fitted parameters, the result is a distribution of stellar parameters that fit the observational data, and from these fitted parameters we integrated the corresponding semi-empirical flux distribution.We therefore obtained a distribution of F bol for each star, and from these distributions we calculated the medians and the symmetric 68% confidence intervals, and report half of the latter as the uncertainty.
Two examples of the data and the best-fitted model SED are shown in Fig. 4, left and right panels.The left is an example of a star with many observational points (in this case HIP103598), while the right panel shows an example where relatively few data points are available; in this case HIP50564.The lower panels show the distribution of the fitted F bol and the individual χ 2 values from the 400 Monte Carlo simulations, along with the value of the adopted median and 16 and 84 percentile confidence levels (dashed lines).We defined the uncertainty as the half of the distance between the upper and lower confidence levels.
The distribution of F bol and relative uncertainties is shown in Fig. 5.The histogram of uncertainties shows a clear peak in the first bin corresponding to uncertainties lower than 0.5%.The relative uncertainties have a median value of 1.4%, and they are lower than 10% except for two stars, namely HIP8837 (ψ Phe) and HIP14135 (α Cet).These two M giants combine a low T eff and a lack of spectrophotometric data which make their SED poorly constrained, resulting in a relative uncertainty of about 21% and 18%, respectively.They had uncertain F bol in Paper I as well.We note that the stars with uncertainties larger than 4% have their SED made of broad-band photometry only, while the majority of stars have spectrophotometry from Gaia and/or Pulkovo, resulting in a very precise F bol determination.
In this procedure to determine F bol , we use log g and metallicity from the PASTEL catalogue, a compilation of literature work.We have evaluated the impact of not knowing precisely these parameters.To do so, we made two tests.One test is to adopt a large uncertainty of 0.15 on both log g and [Fe/H] inducing a different distribution of the fitted F bol from the 400 Monte Carlo simulations.The other test is to change the literature values of log g and [Fe/H] by an amount of 0.15 dex, in the eight possible configurations, for seven stars selected to cover the parameter space.In this test, the largest effect (<1%) is reached when adding 0.15 dex to [Fe/H] for the hottest stars.Varying log g has more of an effect on the coolest stars.When combining the variations of log g and [Fe/H] the effect remains at the level of 1% for the coolest and the hottest stars.Interestingly the most metal-poor star chosen for that test, HIP48152, is less affected by a change of log g and [Fe/H].In the other test, enlarging the log g and [Fe/H] uncertainties in the Monte-Carlo simulations also has a low impact on the derived value of F bol .We note four stars with F bol changed by 1-2%, while 90% of the sample changes by less than 0.5%.We conclude that our procedure weakly depends on the input values of log g and [Fe/H].A change of 1% in F bol induces a change of 0.2% on T eff .However a more rigorous treatment will be performed through iterations once the spectroscopic analysis of the targets will be performed to derive [Fe/H] homogeneously (Paper VIII in preparation).This will lead to self-consistent parameters.
In Paper I, the F bol values of the V1 stars were compiled from the literature and therefore not as homogeneous as here.This is another important improvement of the GBS V3, in addition to the larger number of stars.We still have a good agreement between V1 and V3, with a slight offset of 1.4%, and a typical 557 dispersion of 2.1% (median absolute deviation, MAD).

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Our approach is similar to that of Boyajian et al. (2013)

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In addition they did not take photometric uncertainties into ac-583 count for the fit, while we do.Baines et al. (2018) determine a high extinction for some stars which seems correlated with a larger positive offset.HIP47431 and HIP90344 are the most extreme cases with A V =0.7 mag and A V =0.54 mag respectively in Baines et al. (2018) while we get A V =0.02 mag and A V =0.03 mag from the 3D maps of Vergely et al. (2022), leading to a difference of 50% and 37% on F bol (HIP47431 is not shown in Fig. 6).Considering the 24 stars in common, Baines et al. (2018) find F bol higher than us by 7.6% (median) with a dispersion of 6.6% (MAD).
We also compared our F bol determinations to those of González Hernández & Bonifacio (2009) who implemented the infrared flux method (IRFM) based on 2MASS magnitudes (see Fig. 6).The 61 stars in common generally agree well with an offset less than 1% and a dispersion of 3.9% (MAD).The extinction is low for the majority of these nearby stars.
Finally, we also made a comparison with the catalogue of empirical bolometric fluxes and angular diameters of 1.6 million Tycho-2 stars built by Stevens et al. (2017) which has 119 stars in common with us.This work is based on the flux-colour relations of Casagrande et al. (2010) with T eff and A V being determined separately in an iterative way.Their F bol are globally larger than ours by 4.9%, with a dispersion of 7.5% (MAD).Similarly to the tendency observed in the comparison with Baines et al. (2018), the larger differences correspond to stars with the largest values of A V in Stevens et al. (2017) which significantly differ from our lower extinctions.Eight stars do not appear in Fig. 6, given    9) are the metal-poor benchmark stars HIP57939 and HIP76976 (HD103095 and HD140283).Our new T eff values are about 400 K and 250 K higher than in Paper I, where their sub-mas angular diameters were quoted as very uncertain.Both stars have been remeasured by Karovicova et al. (2018Karovicova et al. ( , 2020) ) leading to more precise θ LD and higher T eff .Our determination for HD103095 (T eff =5235±18 K) is larger by 61 K than that of Karovicova et al. (2020).Since we use their determination of θ LD , the difference is only due to F bol .As noted in Sect.3.1, the angular diameter of HD103095 measured by Karovicova et al. (2020) from the combination of two instruments is very reliable.For HD140283 we find T eff =5788±45 K, lower by 4 K than their value.Three other stars differ by 2-3% from Paper I: ψ Phe, 61 Cyg B, and γ Sge.Only γ Sge has a new angular diameter measured by Baines et al. (2021), while for the other ones we used the same θ LD as in Paper I, indicating that the difference comes from the new determination of F bol , which we expect to be more accurate than the previous determination.
Table 2 and Fig. 9 exhibit larger discrepancies in the comparison to Boyajian et al. (2013) with an offset of 59 K and a scatter of 58 K.We note that we have 66 stars in common but 82 measurements since Boyajian et al. (2013) provide a compilation of their own θ LD together with other values from the literature (we removed discrepant values quoted by them for HD146233 and HD185395).Among the stars that differ by more than 300 K, we have again HD103095 which is the largest outlier.As explained above, the recent θ LD determination by Karovicova et al. (2020) gives a higher T eff which is in better agreement with our value for that star.For HIP61317 Boyajian et al. (2013) give two values of T eff , only one being in significant disagreement with ours.For HIP89348 our values of F bol and θ LD (the latter adopted from Ligi et al. 2016) are larger and smaller, respectively, by ∼10% than those of Boyajian et al. (2013), resulting in a significantly different T eff .Our value of T eff =6569±69 K seems however more consistent with spectroscopic values listed in the PASTEL catalogue than their lower value of T eff =6221±39 K.
The comparison to van Belle et al. (2021) gives an offset of 61 K, this time our values being lower, with a dispersion of 77 K.This relies on 17 giants in common.These large differences could partly be due to disagreement in extinction values for some stars.We note four stars (HIP7607, HIP111944, HIP74666, HIP3031) that van Belle et al. ( 2021) found significantly reddened (A V from about 0.15 to 0.30 mag) while our A V determinations are below 0.05 mag.This possibly explains the T eff differences from 150 K to 220 K. On the other We note that we use the same determination of angular diameter as in the literature for some of the stars.Hence, the comparison data sets are not completely independent from ours.

Surface gravity
We determined the surface gravity log g with the fundamental relation expressed as: where M/M and R/R are the mass and radius of the star in solar units.For the Sun, we adopt for the surface gravity log g = 4.4380 ± 0.0002 dex4 determined in Paper I. The linear radius of each star is deduced from its angular diameter (see Sect. 3.1) and its distance is inferred from its parallax (see be-   With parallaxes π and θ LD we computed linear radii R and with θ LD and π expressed in mas, and F bol in 732 The radii of the GBS V3 span 0.6 to ∼140 R (see Fig. 11).

733
The luminosities span 0.08 to nearly 6000 L (see Fig. 12).were available: 5 https://www.iau.org/static/resolutions/IAU2015_English.pdf where we adopt the fundamental T eff determined in Sect. 3 and the solar parameters as in Paper I: ∆ν = 135.229± 0.003 µHz, ν max, = 3160 ± 40 µHz, T eff, = 5771 ± 1K.We have compiled ∆ν and ν max from the literature and found determinations of both parameters for 37 stars.The comparison is shown in Fig. 13.There is a small systematic offset, the fundamental radii being larger than the seismic ones by 0.7%, with a typical dispersion (MAD) of 3.3%.Several stars show discrepancies larger than 10%, up to 22% for HIP92984.There is however an ambiguity about the seismic parameters of HIP92984, measured by Mosser et al. (2009) from CoRoT observations, because Huber et al. ( 2012) did not detect solar-like oscillations.The other discrepant stars have error bars that still give an agreement at the 3σ level.We also note that Sharma et al. (2016) and Hon et al. (2022) proposed some corrections to the scaling relations to obtain a better agreement for giants.It is however out of the scope of this paper to apply such corrections.We retain from this comparison the general good agreement, with no systematics, between our values and seismic ones, at the level of ∼4%.3), and seismic estimations from Eq. ( 5).
Figure 14 shows our derived luminosities compared to those available for 36 stars in the Gaia DR3 Golden Sample of Astrophysical parameters for FGKM stars (Gaia Collaboration et al. 2023a).Gaia luminosities were computed from the parallax, the G magnitude and a bolometric correction (Creevey et al. 2023) and are therefore different from our determinations, although not completely independent.The bolometric corrections depend on the atmospheric parameters in DR3 and could contribute to some differences that we find.The three most luminous stars in common are found brighter by Gaia by more than 10%, up to 30% for HIP70791, known as a horizontal branch star.Only that star shows a discrepancy significantly larger than 3σ.We note six other stars with Gaia luminosities significantly larger than our values, with differences ranging from 5% to 10%.These discrepancies cannot be explained by the extinction that we find negligible for these nine stars.For the other stars, we find luminosities slightly larger than those from Gaia, by 0.35% (median), with a typical dispersion of 0.9% (MAD).

Masses
Masses were computed with the SPInS code (Lebreton & Reese 2020) implemented with the stellar evolutionary tracks from BaSTI (Pietrinferni et al. 2004(Pietrinferni et al. , 2006)), and from STAREVOL (Lagarde et al. 2012(Lagarde et al. , 2017)).We implemented the two grids in order to make comparisons owing to the different behaviour of the tracks in some parts of the HR-diagram (HRD), such as the clump.The determination of log g is less accurate at clump luminosity (logg 2.2) because this is a point in the HR diagram  where the evolutionary tracks of different masses and [Fe/H] overlap.In the following, we detail the main differences between these two grids that may have an impact on the position on the HRD and thus on the mass determination with SPInS.
For BaSTI, we use stellar tracks coming from the noncanonical grid covering a mass range between 0.5 M and 10.0 M and a metallicity range [Fe/H] ∈ [−2.27, +0.40] without αenhancement.This grid takes into account core convective overshooting during the H-burning phase.The overshoot parameter is set to 0.2 for a stellar mass higher than 1.7 M , no overshooting is considered for a mass lower than 1.1 M , and a linear variation is assumed in-between.The solar mixture comes from Grevesse et al. (1993).We tried the α-enhanced tracks ([α/Fe]=+0.4) for metal-poor stars ([Fe/H]<-0.70 dex) leading to masses higher by 30% on average.However, as explained later, we got a wrong mass for µ Cas, the only metal-poor binary with a reliable dynamical mass.This convinced us to adopt the tracks without αenhancement for the whole sample.
For STAREVOL, the stellar grid covers a mass range between 0.6 and 6.0 M and a metallicity range [Fe/H] ∈ [−2.14, +0.51] without α-enhancement, with the exception of [Fe/H]=-2.14and -1.2 where [α/Fe]=+0.3.Except for convection, additional mixing effects such as rotation-induced mixing are not taken into account.The overshoot parameter is set to 0.05 or 0.10 for stars with masses below or above 2.0 M , respectively; no overshooting is considered for masses lower than 1.1 M .The stellar grid is constructed using the solar mixture coming from Asplund et al. (2009).
The Kroupa initial mass function (Kroupa 2001;Kroupa et al. 2013, IMF) was used as a prior, as well as a truncated uniform star formation rate between 0 and 13.8 Gyr, that is, roughly the age of the Universe.The stellar properties used as an input to SPInS are: (1) our fundamental T eff determinations (Sect.3); (2) luminosities deduced from F bol and parallaxes; (3) metallicities from the literature, and (4) radii deduced from θ LD and parallaxes.Radii are not independent of T eff and luminosities, but  2021) who determined masses of 1.0788±0.0029M and 842 0.9092±0.0025M for the A (HIP71683) and B (HIP71681) 843 components, respectively.The agreement is at the 0.4% level 844 for the STAREVOL mass and 3.4% for the BaSTI mass, for the 845 component A. Both sets of evolutionary tracks give masses that 846 differ by 5% for the B component, in opposite directions.The 847 dynamical mass of the metal-poor ([Fe/H]=-0.83dex) visual bi-848 nary µ Cas (HIP5336) results from an astrometric study with the 849 Hubble Space Telescope by Bond et al. (2020) who determined 850 a value of 0.7440±0.0122M .BaSTI and STAREVOL under-851 estimate it by 4% and 1.3% respectively.Running SPInS with 852 the α-enhanced BaSTI tracks ([α/Fe]=+0.4) for that star led to 853 an overestimation of its mass by 32%.This convinced us not 854 to adopt the α-enhanced tracks for metal-poor stars.Hence, we 855 have opted to exclusively rely on the BaSTI tracks that do not 856 incorporate any alpha-enrichment.This underscores the impor-857 tance of presenting mass values obtained from both BaSTI and 858 STAREVOL tracks, since it offers an understanding of the in-859 herent errors linked to relying solely on a single stellar evolution 860 model.The orbit of Procyon (HIP37279) based on Hubble Space 861 Telescope astrometry (Bond et al. 2015(Bond et al. , 2018)), yields a dynam-862 ical mass of 1.478±0.012M .The BaSTI mass differs by 0.9% 863 while the STAREVOL mass is lower by 2.6%.There is there-864 fore a satisfactory agreement between the SPInS masses and the 865 lutionary tracks, considering the few constraints we use with the We estimated seismic masses using the following scaling re-  We also compared the two sets of SPInS masses to masses 885 from the literature, based on different evolutionary tracks and 886 methods.Figure 18 shows comparisons to masses from Paper I, Baines et al. (2018) and Boyajian et al. (2013).In Paper I masses were determined by visual interpolation in two grids, the Padova grid (Bertelli et al. 2008(Bertelli et al. , 2009) ) and the Yonsei-Yale grid (Yi et al. 2003;Demarque et al. 2004), the adopted value being the average of the two.Baines et al. (2018)  Figure 19 shows our derived masses compared to those available for 30 stars in the Gaia DR3 Golden Sample of Astrophysical parameters for FGKM stars (Gaia Collaboration et al. 2023a).Gaia masses were derived by comparing Gaia photometric effec-tive temperatures and Gaia luminosities to BaSTI solar metallicity stellar evolution models (Hidalgo et al. 2018;Creevey et al. 2023), and are therefore similar to our determinations.We find an excellent agreement between our two sets of masses and the Gaia ones, except for the three stars in common with the highest masses ( 1.5 M ), and one outlier within the STAREVOL set.From the above comparisons, there is no strong evidence that one set of evolutionary tracks is better than the other one.Therefore, we provide the two masses and their uncertainties in the catalogue available at the CDS.
In this procedure to determine masses we need metallicities However, due to the dependency of log g on the logarithm of mass in Eq. ( 2), in the worst cases where the mass is changed by 30%, the impact on log g is limited to 0.11 dex and up to 0.5 dex for the most critical cases.

Assessment of log g
We computed the fundamental log g of each star by applying Eq.
(2) with the values of mass from SPInS, with both evolutionary tracks BaSTI and STAREVOL, and the radius deduced from θ LD , with the propagation of their uncertainties.We consider here the 201 stars of the sample.The resulting uncertainties on log g span from 0.004 to 0.13 (BaSTI) and 0.23 dex (STAREVOL), with a median value of 0.02 dex. Figure 20 shows the compar-956 ison of log g determinations, using the mass from SPInS with 957 BaSTI or STAREVOL.The agreement is excellent for dwarfs 958 with log g>4.Below that value, log g from STAREVOL is sys-959 tematically larger by 0.06 dex than log g from BaSTI, with an 960 exception around log g STAREVOL =2.3.Following the comparisons made in the previous sections, 962 for radii and masses, we used the seismic data to determine log g 963 in another and independent way, through the relation that gives 964 log g as a function of the maximum of the power spectrum of 965 oscillation frequencies, ν max , available for 42 stars, and the ef-966 fective temperature: The comparison of seismic and fundamental log g is shown 968 in Fig. 21.For dwarfs, typically log g seismic >3.8 dex, the agree-969 ment is very good except for one star, HIP92984.For this star 970 we find log g=4.48 dex with BaSTI and STAREVOL masses, 971 while the seismic log g is significantly lower, log g=4.23.We 972 have pointed out the ambiguity about the seismic parameters 973 of that star in the previous section.If this star is excluded, the 974 differences between seismic and fundamental log g of dwarfs 975 have a MAD of 0.02 dex.For giant stars, the dispersion is larger 976 (MAD=0.07dex), with log g based on BaSTI lying slightly be-977 low the seismic values (median offset of −0.06 dex), while the 978 log g based on STAREVOL tend to lie above (median offset of 979 0.01 dex).A few outliers, reaching nearly 0.5 dex, correspond 980 to stars with one of the two masses giving a disagreement with 981 the seismic log g but not the other one.From that comparison, 982 we cannot say that the agreement is better with BaSTI or with 983 STAREVOL masses.

984
We also compare our log g determinations with those in Pa-985 per I and in Karovicova et al. (2020Karovicova et al. ( , 2022a,b) ,b) in Fig. 22. Off-986 0:5 1:0 1:5 2:0 2:5 3:0 3:5 4:0 4:5 5:  The final log g distribution and uncertainties are shown in Fig. 23.The bottom panel shows separately the uncertainties for dwarfs (log g>3.8) and giants (log g≤3.8), highlighting the lower precision obtained for giants.The median uncertainty is 0.02 dex for dwarfs and 0.06 dex for giants.While 90% of the dwarfs have an uncertainty lower than 0.05 dex, 90% of the giants have an uncertainty higher than 0.03 dex.

The new set of Gaia FGK benchmark stars
The fundamental T eff and log g determined for the 192 GBS V3 stars with a direct value of θ LD are given in Table A.1 of Appendix A while the nine other stars from V1 and V2 with an indirect θ LD are provided in Table 3.The metallicity from the literature is provided for convenience, but will be redetermined homogeneously in the coming Paper VIII.The full catalogue with all the other parameters determined in this work is available in VizieR.
The Kiel diagram with fundamental T eff and log g is shown in Fig. 24 for the full sample of 192 stars and for a selection of the best stars, with an uncertainty on T eff and log g better than 2% and 0.1 dex, respectively.This selection of 165 stars mainly rejects giants with large uncertainties, as discussed in Sect.4, but still preserves a good distribution across the Kiel diagram.
The metallicity histogram of the GBS V3 is shown in Fig. 25, compared to that of V1 (only considering stars with a direct θ LD ), highlighting a number of new metal-poor stars.This is however more evident in the interval −1.0 <[Fe/H]< −0.5 than below [Fe/H]=−1.0.There were four stars in GBS V1 with −1.0 <[Fe/H]< −0.5, a number increased to 14 in GBS V3.Four    One of the main purposes of the GBS is to calibrate or 1037 validate atmospheric parameters from spectroscopy.We there-1038 fore checked spectroscopic T eff and log g available in different 1039 sources, using the subset of 165 most reliable GBS.
1040 Fig. 26 compares our fundamental determinations with those available in the PASTEL catalogue, based on high-resolution, high signal-to-noise spectroscopy.Overall, the agreement on T eff is good with a dispersion of MAD=54 K and a slight offset of 12 K (median), the spectroscopic T eff being larger.Two extreme outliers have differences larger than 400 K.For HIP86614 we suspect an uncertain angular diameter given its noisy squared visibility curve in Boyajian et al. (2012b) while for HIP108535 the only spectroscopic T eff in PASTEL is dubious.For that star we note a good agreement with the determination by Prugniel et al. (2011) based on a medium resolution spectrum.
Concerning log g we can see three regimes of precision, corresponding to dwarfs, clump giants, and cooler giants, with an increasing dispersion.The dispersion among dwarfs is 0.04 dex (MAD).It rises to 0.1 dex among clump giants (2.0<log g<3.5) with no offset, while for red giants there is a tendency of spectroscopic log g to be larger than the fundamental ones by 0.16 dex (median offset) with a significant dispersion of 0.2 dex (MAD).The GBS can therefore be used to better understand and correct the spectroscopic gravities of evolved stars.
Focusing on the best studied stars we selected in the PAS-TEL catalogue the stars which are included in at least 15 spectroscopic studies at high-resolution and high signal-to-noise ratio since 1990.The resulting 16 stars are all dwarfs or subgiants, with some of them also in common with Paper I. In general there is a good agreement, within our uncertainties and the standard deviation from the literature values.Three stars, HIP14954, HIP57939 and HIP8159, exhibit a significant difference in T eff , larger than 150 K. HIP14954 (94 Cet) has been very much studied, with 41 spectroscopic determinations of T eff , likely because of its exoplanet discovered in 2000 (Queloz et al. 2001).The literature values range from 5916 K to 6424 K with a mean of 6176 K and a standard deviation of 84 K. Our determination is lower, T eff =5912±59 K, but still in agreement with the coolest spectroscopic determinations.Our fundamental value is in a very good agreement with that of Boyajian et al. (2013), T eff =5916±98 K, independent from ours since we use the angular diameter from Ligi et al. (2016).It would be important to better understand why spectroscopy gives a higher T eff for that star because it has implications on the parameters of its exoplanet.HIP8159 (109 Psc) also hosts an exoplanet and has several recent T eff from high-resolution spectroscopy ranging between 5560 K and 5711 K.The fundamental determinations,   from Boyajian et al. (2013) and from us (T eff =5438±61 K) based on the same θ LD , are cooler than the mean spectroscopic value by ∼200 K.This discrepancy requires further investigation.HIP57939 (HD103095) has 57 spectroscopic determinations of T eff after 1990, ranging from 4500 K to 5250 K with a mean of 5057 K and a standard deviation of 18 K.Our fundamental determination T eff =5235±18 K is in agreement with the hottest spectroscopic determinations, for example, by Luck & Heiter (2005).
We also checked atmospheric parameters massively determined by large spectroscopic surveys against our fundamental determinations of the best GBS.We considered APOGEE DR17 (Majewski et al. 2017;Abdurro'uf et al. 2022), the Gaia-ESO survey (Randich et al. 2022;Gilmore et al. 2022) and GALAH DR3 (Buder et al. 2021) in the comparisons shown in Fig. 27.
Table 5 gives the median offsets and corresponding MAD for dwarfs and giants separately.Although GALAH and Gaia-ESO have less stars in common than APOGEE, we see the same trends in the three surveys.Their T eff and log g for dwarfs are smaller on average than the fundamental ones, and vice-versa for the giants.These trends are worth to be investigated and better understood.
Finally we also assessed the photometric and spectroscopic T eff and log g of the Gaia DR3 Golden Sample of Astrophysical Table 5. Median difference (MED) and median absolute deviation (MAD) between our fundamental determinations of T eff and log g and the spectroscopic ones from surveys (survey results minus our results), for N stars in common.At each stage of the compilation and determination of the parameters, we evaluated the uncertainties that we aimed to keep at the 1-2% level.Our results were assessed by comparing them to other determinations of a similar accuracy available in the literature.In general, the comparison with literature data is satisfactory, with differences not exceeding 4%.We can explain most of the outliers.We also determined seismic radii, masses, and surface gravities for comparisons, using scaling relations and seismic parameters available for ∼40 stars.They show a good agreement for dwarfs but a trend in masses outside the 1-1.5M range.From the different comparisons, we are confident that our uncertainties in T eff are reliable.We reached the expected 1-2% level in T eff .For log g only dwarfs have such a level of accuracy.Uncertainties for giants are larger and reflect the difficulty to obtain reliable masses for them from evolutionary tracks.

Conclusion
The T eff and log g presented here will be updated.Two steps of our determination process, F bol and masses, depend on [Fe/H] which we took from the literature.This is the subject of the upcoming paper VIII to determine abundances of the GBS-V3 from a large collection of high-quality spectra.Some iterations will be needed to adjust T eff , log g , and [Fe/H] in a self-consistent way.In the meantime, we have evaluated the impact of using nonhomogeneous metallicities through tests in which we modified the values of [Fe/H] and uncertainties by 0.15 dex in input of the SED fitting and of SPInS.We found that it has a negligible impact on T eff , and also on log g for most of the stars, although a few giants have their mass affected by more than 30% inducing a change of log g by 0.11 to 0.5 dex.
In order to use the GBS V3 for calibration or validation of atmospheric parameters, we recommend that users select the 165 stars with uncertainties on T eff and log g lower than 2% and 0.1 dex, respectively.We have used this subsample to assess T eff and log g obtained by high-resolution and high signal-to-noise spectroscopy (PASTEL catalogue), by medium-resolution spectroscopy (APOGEE, GALAH, and Gaia-ESO surveys), and by Gaia photometry and spectroscopy.This has revealed some issues that need to be investigated to improve the future releases.
Due to the lack of metal-poor stars in the solar neighbourhood, the GBS V3 do not yet cover the metallicity range in a uniform way.We still lack angular diameters for stars with [Fe/H]< −1.0, which are important targets in galactic archeology and stellar physics.Interferometric measurements are still limited to stars brighter than V∼8, and larger than θ LD 0.2 mas.There are, however, metal-poor candidates that are bright and large enough to fulfil these criteria.They could be observed with powerful interferometers, such as the new SPICA instrument on the CHARA array (Mourard et al. 2022) expected to provide an estimation of the stellar radius of such stars to a 1% precision.It would also be useful to remeasure, either in part or entirely, the GBS with θ LD <1.2 mas, which show a large dispersion of the current measurements, exceeding the quoted uncertainties.

2
fort aimed at deciphering the history of our Galaxy based on 3 large samples of stars observed by spectroscopic surveys.This 4 has stimulated the development of efficient methodologies for 5 the massive determination of atmospheric parameters (APs).In 6 particular, the recent Gaia Data Release 3 (Gaia Collaboration 7 et al. 2023c, Gaia DR3) just delivered T eff , log g , and [Fe/H] for 8 millions of stars metric one, tend to reject metal-poor stars.The only star with [Fe/H]< −1.0 in Salsi et al. (2020)'s sample is the well-known benchmark star HIP76976 (HD 140283), part of GBS V1, which has [Fe/H]=−2.36±0.10 in Paper IV.

Fig. 2 .
Fig. 2. Difference between θ LD adopted for this work and other values in JMDC fulfilling the selection criteria by Salsi et al. (2020).

A
and B are not resolved in the 2MASS catalog 2 , and the magnitudes given for α Cen A contain actually the combined flux of both components.We therefore used J, H, and K magnitudes fromEngels et al. (1981), which are given for each component separately, and converted them to flux values using the VOSA tool.An interesting new feature of the latest VOSA version (July 2022 update) is to provide synthetic photometry based on Gaia DR3 BP/RP spectra analysed with the GaiaXPy tool (De Angeli et al. 2023; Gaia Collaboration et al. 2023b).We therefore collected through VOSA the synthetic photometry from Gaia which is provided in 13 passbands corresponding to the filters of the Hubble Space Telescope, Sloan Digital Sky Survey, PanSTARRS1 and Johnson UBVRI systems.Also from Gaia BP/RP spectra and GaiaXPy, VOSA computes fluxes in the 65 bands of the OAJ/J-PAS and OAJ/J-PLUS surveys.However we noticed that a small fraction of the Gaia synthetic photometry was affected by saturation, causing the corresponding SED to be deformed.We had to remove the Gaia spectrophometry, totally or partially, for about thirty bright stars with G 4. Finally we added to the compilation the fluxes in the range 320-1080 nm from the Pulkovo spectrophotometric catalogue (Alekseeva et al. 1996), adopting a homogeneous uncertainty of 1% for each value of flux (this value allowed us to give these data an appropriate weighting in our analysis).The Pulkovo catalogue provides 167 or 305 flux values, depending on the star.The details of the num-An illustration of the obtained SEDs is given for two stars in 416 Fig. 4 in Sect.3.4.

Fig. 3 .
Fig. 3. Extinction A V deduced from 3D maps of Vergely et al. (2022) as a function of distance, for the 201 targets.
457sampled grid of 1221 points, with a mean resolution of 100 nm 458 in the UV and 200 nm in the visible.Beyond 10 000 nm the res-459 olution is 20 000 nm and to avoid issues with numerical integra-460 tion we interpolate on a log scale before performing the integra-461 tion.A Levenberg-Marquardt minimisation algorithm finds the 462 optimal template that fits the observed flux points.F bol is then 463 calculated by integrating the optimal fitted spectrum.Recent im-464 provements of the method include the weighting of the fluxes 465 and the determination of F bol uncertainties through Monte-Carlo 466 simulations.467 The parameters of the model are the atmospheric parameters: 468 T eff , log g, [Fe/H], the extinction, and the scaling factor (stel-469 lar radius scaled according to the distance).We used the atmo-470 spheric parameters from PASTEL to initialise the minimisation 471 and the extinction from Sect.3.3.To account for extinction in 472 our method we implemented the IDL routine ccm_unred 3 which 473 dereddens theoretical fluxes, and requires colour excess on in-474 put.To convert extinction to colour excess we adopted R 0 = 3.1.475 Most of these stars are nearby and as such have little or no ex-476 tinction.In order to be complete in our analysis, in the catalogue 477 available at the CDS we also provide F bol for the full sample of 478 stars by assuming zero extinction.Note that for stars with ex-479 tinction close to 0 mag, the fitted F bol is not necessarily smaller 480 when assuming A V = 0.0 mag, because A V impacts the shape 481 more than the height of the SED over the full spectral range.482 All of the above parameters can be fitted, but in practice due 483 to degeneracies between the parameters, the T eff and the scaling 484 factor are the only free parameters, while log g, [Fe/H], and the 485 extinction are fixed each time a minimisation is performed.In 486 order to include the impact of the uncertainties of the parameters 487 log g and [Fe/H], and of the fluxes, we performed a bootstrapped-488 based method where we (a) perturbed these parameters by their 489 uncertainty multiplied by a random number drawn from a Gaus-490 sian distribution, and (b) perturbed the fluxes by their uncertain-491

Fig. 4 .
Fig. 4. Example of fits of the observed (reddened) data to the (reddened) semi-empirical spectra for HIP 103598 (left) and HIP 50564 (right).The bolometric flux is calculated by integrating the un-reddened spectrum.The bottom panels illustrate the distribution of χ 2 R versus F bol for the 400 simulations for the two stars with the 16 th , 50 th (median), and 84 th percentiles indicated by the dashed lines.The continuous line is the derived F bol without considering the simulations.
559andBaines et al. (2018)  who collected broadband photomet-560 ric measurements available in the literature, extended by spec-561 trophotometry when available.They also applied the SED fitting 562 method with reference templates taken from the library of Pick-563 les (1998) which is made of observed spectra, whereas we used 564 a hybrid library of synthetic stellar spectra calibrated from ob-565 servations(Lejeune et al. 1997).Another difference comes from 566 the Gaia spectrophotometry recently made available, which con-567 strains very well the SED shape in the optical range.We have 66 568 stars in common withBoyajian et al. (2013) and 24 with Baines 569et al.(2018).The F bol comparison is shown in Fig.6.The agree-570 ment withBoyajian et al. (2013) is very good, with differences 571 within 10%.On average our F bol values are higher than their 572 values by 3.3%, with a typical dispersion of 2.5% (MAD).The 573 offset does not seem correlated with extinction which is lower 574 than 0.03 mag for the stars in common according to our estima-575 tions, and that they have not considered given the close distance 576 of the stars.It is likely that the small offset observed between our 577 F bol determinations and those of Boyajian et al. (2013) is related 578 to their use of magnitudes from photometric catalogues, with a 579 maximum of 17 values per star and fewer than 12 values in most 580 cases, while we have typically ten times more flux values, mostly 581 from Gaia spectrophotometry, providing SEDs of better quality.

Fig. 6 .Fig. 7 .Fig. 8 .
Fig. 6.Comparison of F bol obtained in this work with literature.The colour code relates to the extinction.Several extreme outliers are out of the figure boundaries but they are discussed in the text.

734
Solar-like oscillations provide robust constraints to the ra-735 dius of G and K dwarfs and giants (Chaplin & Miglio 2013), 736 giving us an opportunity to compare our determinations with 737 others obtained in a different way.We estimated seismic radii 738 using the following scaling relation (e.g.Miglio 2012) when the 739 asteroseismic parameters, the so-called large frequency separa-740 tion ∆ν and the frequency of maximum oscillation power ν max , 741

Fig. 14 .
Fig. 14.Luminosity difference between our determinations L from F bol and distance, and Gaia DR3 estimations based on G magnitudes and bolometric corrections for 36 stars in common in the Golden Sample of Astrophysical Parameters (Gaia Collaboration et al. 2023a).

Fig. 16 .
Fig. 16.Comparison of masses determined with SPInS (red dots for BaSTI, blue open squares for STAREVOL) to dynamical masses for µ Cas, α Cen B, α Cen A and Procyon, ordered by increasing mass.
871 lation (e.g.Miglio 2012) for the 37 stars having a determination 872 of the asteroseismic parameter ∆ν available in the literature (see 873 Sect.4.1 for the solar values) the linear radius computed in Sect.4.1.The re-875 sulting comparison is shown in Fig. 17.Although the agreement 876 between SPInS and seismic masses is good in general in the 877 range 1-1.5 M , there is a trend in the sense that SPInS tends 878 to overestimate masses smaller than 1 M and to underestimate 879 those larger than 1.5 M .This is true for both sets of evolutionary 880 tracks, with more outliers with STAREVOL.However, seismic 881 masses may not necessarily be more accurate than those deduced 882 from evolutionary tracks, given that the range of validity of the 883 scaling relation is not yet clear (e.g.Sharma et al. 2016).

Fig. 18 .
Fig. 18.Comparison of masses determined with SPInS (red dots for BaSTI, blue open squares for STAREVOL) to those available in the literature, also based on evolutionary tracks.

Fig. 19 .
Fig. 19.Comparison between masses from SPInS (red dots for BaSTI, blue open squares for STAREVOL) and masses from the Gaia Golden Sample of Astrophysical Parameters (Gaia Collaboration et al. 2023a).
as input for SPInS.We have used [Fe/H] values from the literature which are not homogeneous and therefore we have evaluated their impact on the resulting masses.We made two tests similar to those made for F bol .One test is to adopt a large uncertainty of 0.15 on [Fe/H] for all the stars, much larger than the original ones.The other test is to add or subtract 0.15 dex to the literature values of [Fe/H] for seven stars selected to cover the parameter space.Enlarging the metallicity uncertainty to 0.15 dex affects mainly the clump giants.Based on the BASTI tracks, only five stars in our sample have their mass affected by more than 30%, and only eight stars if we consider the STAREVOL tracks.The most critical stars are not common from one set to the other.This reinforces the interest of considering the masses computed by the two sets of stellar evolution models.Such differences could be explained by different inputs in the computation of the evolutionary tracks (e.g.mass loss, atmosphere models, etc.) which change the position in the HRD.It should be noted that at least 90% of the stars in our sample experience a mass variation less than 10%, while three quarters of the sample remain below 5%, whatever the set of stellar models taken into account.Changing the value of [Fe/H] by ±0.15 dex for seven stars leads to a similar conclusion: the dwarfs are not affected, whatever their metallicity, while changes occur among giants.

Fig. 20 .
Fig. 20.Comparison of log g determinations, using masses from SPInS with BaSTI or STAREVOL.The area between the two dashed lines indicates an agreement within 0.1 dex.

Fig. 21 .Fig. 22 .
Fig. 21.Comparison of our fundamental values of log g to those determined from ν max and our fundamental T eff .Red dots for SPInS masses using BaSTI models, blue open squares for STAREVOL models.

Fig. 23 .
Fig. 23.Histogram of log g (top panel) and uncertainty (bottom panel).The bottom panel shows the uncertainties for dwarfs (red) and giants (blue).

Fig. 24 .
Fig. 24.Kiel diagram with fundamental T eff and log g.The colour scale is related to metallicities from the literature.The left panel shows the full sample of 192 stars while the right panel shows the stars with uncertainties on T eff and log g better than 2% and 0.1 dex respectively.

1033
number of stars with [Fe/H]< −1.0.The fundamental T eff and 1034 log g for these eight GBS V3 stars are presented in Table 4 to-1035 gether with the values from Paper I for the four stars in common.

Fig. 26 .
Fig. 26.Comparison of our fundamental values of T eff (top panel) and log g (bottom panel) to spectroscopic ones available in the PASTEL catalogue.

Fig. 28 .
Fig. 28.Comparison of T eff and log g from this work with the photometric (red dots) and spectroscopic (blue squares) ones from the Gaia DR3 Golden Sample of Astrophysical Parameters (Gaia Collaboration et al. 2023a).

1119
Large spectroscopic surveys usually calibrate or validate their 1120 determinations of atmospheric parameters using reference stars.1121 Ideally they should adopt a common T eff and log g scale in or-1122 der to minimise systematic differences in abundances provided 1123 by different instruments and pipelines.The GBS are intended to 1124 provide such an anchor to the fundamental T eff and log g.GBS 1125 can also help one understand any shortcomings in the stellar 1126 models.1127 We have presented determinations of fundamental T eff and 1128 log g, based on the Stefan-Boltzmann law and Newton's law of 1129 gravitation, for the third version of the GBS.Compared to the 1130 previous V1 and V2 versions, a significant improvement is the 1131 larger number of stars, 192 instead of ∼40, resulting from our 1132 systematic search of high-quality angular diameters based on in-1133 terferometric measurements.More accurate log g were obtained 1134 thanks to the higher precision parallaxes that mostly come from 1135 Gaia DR3, while the improved T eff are in part a result of the 1136 homogenous photometric data from Gaia DR3, which feed into 1137 the F bol determination.We note that F bol are now more precise 1138 and homogeneous owing to the methodology of SED fitting ap-1139 plied to a combination of photometric and spectrophotometric 1140 data including measurements made on BP/RP spectra from Gaia 1141 DR3.Better F bol also result from the adopted extinction values 1142 deduced from state-of-the-art 3D maps of the solar neighbour-1143 hood.The most difficult part comes from the determination of 1144 masses from evolutionary tracks.We have shown that using two 1145 different grids can lead to differences of up to more than 50% 1146 in masses, giving systematic offsets of about 0.06 dex in log g 1147 among giants.1148

Table 2 .
Median difference (MED) and median absolute deviation (MAD) between direct determinations of T eff from the literature and our values (literature minus this work), for N stars in common.

Table 2 .
Karovicova et al. (2020kably low (MAD 30 K) for the comparison to Paper I andKarovicova et al. (2020Karovicova et al. ( ,  2022a,b),b), our determinations being larger by 26 K and 39 K respectively.The agreement is therefore at the 1% level in general.The two outliers in the comparison to Paper I (upper left panel of Fig.

Table 3 .
T eff and log g determined in this work for stars from V1 and V2 with an indirect value of θ LD .[Fe/H] from the literature is given for indication.

Table 4 .
Fundamental T eff and log g for the eight GBS V3 with [Fe/H]< −1.0 (from the literature), and comparison to the Paper I values, when available.
Gaia Collaboration et al. 2023a) with the best GBS, 1108 as shown in Fig.28.Photometric T eff are lower than fundamental 1109 ones by 58 K, while the offset of the spectroscopic T eff is negli-1110 gible (6 K).For log g there is an excellent agreement of the pho-1111 tometric values with median offset of −0.03 dex and a dispersion 1112 (MAD) of 0.06 dex.Spectroscopic log g, corrected as suggested 1113 Recio-Blanco et al. (2023)3), are found smaller than the funda-1114 mental values by 0.06 dex, with a dispersion of 0.15 dex.These 1115 photometric and spectroscopic parameters respectively, mainly 1117 dwarfs.

Table A .
1. Fundamental T eff and log g and their uncertainties determined in this work for the 192 GBS V3, with [Fe/H] from the literature and the θ LD adopted measurement.